cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

A222180 Total number of ON states after n generations of cellular automaton based on pentagons. Also P-toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 6, 16, 26, 36, 56, 86, 106, 116, 136, 176, 216, 246, 296, 366, 406, 416, 436, 476, 536, 616
Offset: 0

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Author

Omar E. Pol, Mar 15 2013

Keywords

Comments

Analog of A161644, A147562 and A151723, but here we are working without a lattice. Each regular pentagon has five virtual neighbors. Overlapping are prohibited. The sequence gives the number of pentagons in the structure after n-th stage. A222181 (the first differences) gives the number of pentagons added at n-th stage.
Also this is a P-toothpick sequence since every pentagon can be replaced by a P-toothpick which is formed by five toothpicks as a five-pointed star. Note that each toothpick can be represented as an apothem or as a radius of a pentagon. In both types of structures the number of toothpicks after n-th stage is equal to 5*a(n).

Links

Crossrefs

Cf. A139250, A147562, A151723, A152968, A161330, A161644, A182632, A182840, A222172, A222176, A222181.

Formula

a(n) = 6 + 10*A222172(n-2), n >= 2. - Omar E. Pol, Nov 24 2013

Extensions

Name improved by Omar E. Pol, Nov 24 2013

A182617 Number of toothpicks in a toothpick spiral around n cells on hexagonal net.

Original entry on oeis.org

0, 5, 9, 12, 15, 18, 21, 23, 26, 29, 31, 34, 36, 39, 41, 44, 46, 49, 51, 53, 56, 58, 61, 63, 65, 68, 70, 72, 75, 77, 79, 82, 84, 86, 89, 91, 93, 95, 98, 100
Offset: 0

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Author

Omar E. Pol, Dec 13 2010

Keywords

Comments

The toothpick spiral contains n hexagonal "ON" cells that are connected without holes. A hexagonal cell is "ON" if the hexagon has 6 vertices that are covered by the toothpicks.
Attempt of an explanation: in the hexagonal grid, we can pick any of the hexagons as a center, and then define a ring of 6 first neighbors (hexagons adjacent to the center), then define a ring of 12 second neighbors (hexagons adjacent to any of the first ring) and so on. The current sequence describes a self-avoiding walk which starts in a spiral around the center hexagon, which covers 5 edges. The walk then takes one step to reach the rim of the first ring and travels once around this ring until it reaches a point where self-avoidance stops it. It then takes one step to reach the rim of the second ring and walks around that one, etc. Imagine that on each edge we place a toothpick if it's on the path, and interrupt counting the total number of toothpicks each time one of the hexagons has six vertices covered. The total number of toothpicks after n-th stage define this sequence. Note that, except from the last sentence, this comment is a copy from R. J. Mathar's comment in A182618 (Dec 13 2010). - Omar E. Pol, Sep 15 2013

Examples

			On the infinite hexagonal grid we start at stage 0 with no toothpicks, so a(0) = 0.
At stage 1 we place 5 toothpicks on the edges of the first hexagonal cell, so a(1) = 5.
At stage 2, from the last exposed endpoint, we place 4 other toothpicks on the edges of the second hexagonal cell, so a(2) = 5 + 4 = 9 because there are 9 toothpicks in the structure.
At stage 3, from the last exposed endpoint, we place 3 other toothpicks on the edges of the third hexagonal cell, so a(3) = 9 + 3 = 12 because there are 12 toothpicks in the spiral.
From _Omar E. Pol_, Sep 14 2013: (Start)
Illustration of initial terms:
.                        _       _         _         _
.              _       _/ \    _/ \_     _/ \_     _/ \_
.  _      _   /  _    /  _    /  _  \   /  _  \   /  _  \
. / \    / \  \ / \   \ / \   \ / \ /   \ / \ /   \ / \ /
.  _/  /  _/  /  _/   /  _/   /  _/     /  _/ \   /  _/ \
.      \_/    \_/     \_/     \_/       \_/  _/   \_/  _/
.                                                    _/
.
.  5     9      12      15       18        21       23
.
(End)

Links

Crossrefs

Cf. A139250, A182618, A182619, A182632, A182840.

Formula

Conjecture: a(n) = 2*n + ceiling(sqrt(12*n - 3)), for n > 0. - Vincenzo Librandi, Sep 20 2017

A182618 Number of new grid points that are covered by the toothpicks added at n-th-stage to the toothpick spiral of A182617.

Original entry on oeis.org

6, 4, 3, 3, 3, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3
Offset: 1

Views

Author

Omar E. Pol, Dec 12 2010

Keywords

Comments

In the toothpick spiral the toothpicks are connected by their endpoints. See A182617 for more information.
Attempt at an explanation, R. J. Mathar, Dec 13 2010: (Start)
In the hexagonal grid, we can pick any of the hexagons as a center, and then define a ring of 6 first neighbors (hexagons adjacent to the center), then define a ring of 12 second neighbors (hexagons adjacent to any of the first ring) and so on. The current sequence describes a self-avoiding walk which starts in a spiral around the center hexagon, which covers 5 edges. The walk then takes one step to reach the rim of the first ring and travels once around this ring until it reaches a point where self-avoidance stops it. It then takes one step to reach the rim of the second ring and walks around that one, etc. Imagine that on each edge we place a toothpick if it's on the path, and interrupt counting the total number of toothpicks each time one of the hexagons has six vertices covered. The first differences of these intermediate totals define the sequence. (End)

Examples

			At stage 1, starting from a node on the hexagonal net, we place 5 toothpicks on 5 edges of the first hexagon, so a(1)= 6 because there are 6 grid points that are covered by the toothpicks.
At stage 2, starting from the last exposed endpoints, we place 4 toothpicks on the edges of the second hexagon, so a(2)=4 because there are new 4 grid points that are covered by the toothpicks.
At stage 3, starting from the last exposed endpoints we place 3 toothpicks on the edges of the third hexagon, so a(3)=3 because there are new 3 grid points covered. Etc.
If written as a triangle, begins:
6,
4,3,3,3,3,2,
3,3,2,3,2,3,2,3,2,3,2,2,
3,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,2,
3,2,2,3,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,2,
3,2,2,2,3,2,2,2,2,3,2,2,2,2,3,2,2,2,2,3,2,2,2,2,3,2,2,2,2,2

Links

Crossrefs

Cf. A121149, A139250, A139251, A182617, A182619, A182632, A182840.
Row n has A008458(n-1) terms. Row sums give A017593.

A183004 Toothpick sequence on square grid with toothpicks connected by their endpoints.

Original entry on oeis.org

0, 1, 5, 11, 19, 27, 43, 65, 81, 89, 105, 129, 153, 185, 241, 303, 335, 343, 359, 383, 407, 439, 495, 559, 599, 631, 687, 759, 839, 959, 1135, 1293, 1357, 1365, 1381, 1405, 1429, 1461, 1517, 1581, 1621, 1653, 1709, 1781
Offset: 0

Views

Author

Omar E. Pol, Mar 27 2011

Keywords

Comments

Rules:
- If n is odd then each new toothpick must lie in vertical direction.
- If n is even then each new toothpick must lie in horizontal direction.
- Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of two toothpicks of new generation.
The sequence gives the number of toothpicks after n stages. A183005 (the first differences) gives the number added at the n-th stage.
The structure is very similar to the structure of A139250 but the mechanism for the connection of toothpicks is different.

Links

Crossrefs

Cf. A139250, A160164, A182840, A183005.

Formula

We start at stage 0 with no toothpicks.
At stage 1, place a single toothpick of length 1 on a square grid, aligned with the y-axis, so a(1)=1. There are two exposed endpoints.
At stage 2, place 4 toothpicks in horizontal position: two new toothpicks touching each exposed endpoint, so a(2)=1+4=5. There are 4 exposed endpoints.
At stage 3, place 6 toothpicks in vertical position, so a(3)=5+6=11.
After 3 stages the toothpick structure has 2 squares and 4 exposed endpoints.

A182836 Toothpick sequence starting at the vertex of the outside corner of an infinite 120-degree wedge on hexagonal net.

Original entry on oeis.org

0, 1, 3, 7, 15, 27, 39, 51, 71, 91, 107
Offset: 0

Views

Author

Omar E. Pol, Dec 12 2010

Keywords

Comments

Corner sequence for the toothpick structure on hexagonal net.
The sequence gives the number of toothpicks after n stages. A182837 (the first differences) gives the number added at the n-th stage. For more information see A182632 and A153006.

Examples

			We start at stage 0 with no toothpicks.
At stage 1 we place a single toothpick touching a vertex of the infinite hexagon, in direction to the center of the hexagon, but on the outside corner, so a(1)=1.
At stage 2 we place 2 toothpicks touching the exposed endpoint of the initial toothpick, so a(2)=1+2=3.
At stage 3 we place 4 toothpicks, so a(3)=3+4=7.
At stage 4 we place 8 toothpicks, so a(4)=7+8=15.
At stage 5 we place 12 toothpicks, so a(5)=15+12=27.
After 5 stages the toothpick structure has 5 hexagons and 6 exposed endpoints.

Links

Crossrefs

Cf. A139250, A153006, A182632, A182634, A182837, A182840.

A182842 a(n) = A182841(n+2)/2.

Original entry on oeis.org

2, 4, 7, 8, 7, 12, 19, 16, 7, 12, 23, 32, 27, 28, 43, 32, 7, 12, 23, 32, 31, 40, 63, 72, 43, 28, 55, 84, 79, 72, 99, 64, 7, 12, 23, 32, 31, 40, 63, 72, 47, 40, 71, 112, 119, 112, 143, 152, 75, 28, 55, 84, 91, 108, 163, 204, 151, 88, 131, 204, 207, 180, 219, 128
Offset: 0

Views

Author

Omar E. Pol, Dec 11 2010

Keywords

Examples

			From _Omar E. Pol_, Nov 01 2014: (Start)
When written as an irregular triangle with row lengths A011782:
2;
4;
7, 8;
7, 12, 19, 16;
7, 12, 23, 32, 27, 28, 43, 32;
7, 12, 23, 32, 31, 40, 63, 72, 43, 28, 55, 84, 79, 72, 99, 64;
7, 12, 23, 32, 31, 40, 63, 72, 47, 40, 71, 112, 119, 112, 143, 152, 75, 28, 55, 84, 91, 108, 163, 204, 151, 88, 131, 204, 207, 180, 219, 128;
The right border gives the even powers of 2, at least up a(2^9-1).
(End)

Links

Crossrefs

Cf. A139250, A139251, A151724, A182633, A182840, A182841.

Extensions

More terms from Olaf Voß, Dec 24 2010
Wiki link added by Olaf Voß, Jan 14 2011

A360512 Total number of edges after n generations in hexagonal graph constructed in first quadrant (see Comments in A360501 for precise definition).

Original entry on oeis.org

0, 1, 2, 4, 8, 13, 19, 26, 34, 44, 54, 66, 78, 93, 107, 124, 140, 160, 178, 200, 220, 245, 267, 294, 318, 348, 374, 406, 434, 469, 499, 536, 568, 608, 642, 684, 720, 765, 803, 850, 890, 940, 982, 1034, 1078, 1133, 1179, 1236, 1284, 1344, 1394, 1456, 1508, 1573, 1627, 1694, 1750
Offset: 0

Views

Author

N. J. A. Sloane, Feb 09 2023

Keywords

Links

Crossrefs

Partial sums of A360501.
Cf. A182632, A182838, A182840.

Formula

G.f. = x*(1+x+x^2+3*x^3+2*x^4+x^5+x^6-x^7)/((1-x)*(1-x^2)*(1-x^4)).
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